University of Oxford

This session will take place live in L1 only and not online on Teams. 

Are you interested in sharing your love of Maths with the next generation of mathematicians, but you don’t know where to start? In this session we will discuss some basic principles and top tips for creating a workshop for students aged 14–16, and get you started on developing your own. There will also be the opportunity to work on this further afterwards and potentially deliver your session as part of the Oxfordshire Maths Masterclasses (for local school students) in Hilary Term. Bring along your favourite bit of maths and a willingness to have a go.

We develop a new class of algorithms for general linear systems and a wide range of eigenvalue problems. These algorithms apply fast randomized sketching to accelerate subspace projection methods.  This approach offers great flexibility in designing the basis for the approximation subspace, which can improve scalability in many computational environments. The resulting algorithms outperform the classic methods with minimal loss of accuracy. For model problems, numerical experiments show large advantages over MATLAB’s optimized routines, including a 100x speedup. 

Joint work with Joel Tropp (Caltech). 

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The additive Schwarz method with vertex-centered patches and a low-order coarse space gives a p-robust solver for FEM discretizations of symmetric and coercive problems. However, for very high polynomial degree it is not feasible to assemble or factorize the matrices for each patch. In this work we introduce a direct solver for separable patch problems that scales to very high polynomial degree on tensor product cells. The solver constructs a tensor product basis that diagonalizes the blocks in the stiffness matrix for the internal degrees of freedom of each individual cell. As a result, the non-zero structure of the cell matrices is that of the graph connecting internal degrees of freedom to their projection onto the facets. In the new basis, the patch problem is as sparse as a low-order finite difference discretization, while having a sparser Cholesky factorization. We can thus afford to assemble and factorize the matrices for the vertex-patch problems, even for very high polynomial degree. In the non-separable case, the method can be applied as a preconditioner by approximating the problem with a separable surrogate. We apply this approach as a relaxation for the displacement block of mixed formulations of incompressible linear elasticity.

Viscous contact problems describe the time evolution of fluid flows in contact with a surface from which they can detach. These type of problems arise in glaciology when, for example, modelling the evolution of the grounding line of a marine ice sheet or the formation of a subglacial cavity. Such problems are generally modelled as a time dependent viscous Stokes flow with a free boundary and contact boundary conditions. Although these applications are of great importance in glaciology, a systematic study of the numerical approximation of viscous contact problems has not been carried out yet. In this talk, I will present some of the challenges that arise when approximating these problems and some of the ideas we have come up with for overcoming them.

Over the last few decades, scientists have conducted extensive research on parallelisation in time, which appears to be a promising way to provide additional parallelism when parallelisation in space saturates before all parallel resources have been used. For the simulations of interest to the Culham Centre of Fusion Energy (CCFE), however, time parallelisation is highly non-trivial, because the exponential divergence of nearby trajectories makes it hard for time-parallel numerical integration to achieve convergence. In this talk we present our results for the convergence analysis of parallel-in-time algorithms on nonlinear problems, focussing on what is widely accepted to be the prototypical parallel-in-time method, the Parareal algorithm. Next, we introduce a new error function to measure convergence based on the maximal Lyapunov exponents, and show how it improves the overall parallel speedup when compared to the traditional check used in the literature. We conclude by mentioning how the above tools can help us design and analyse a novel algorithm for the long-time integration of chaotic systems that uses time-parallel algorithms as a sub-procedure.

The solution of systems of linear(ized) equations lies at the heart of many problems in Scientific Computing. In particular for large systems, iterative methods are a primary approach. For many symmetric (or self-adjoint) systems, there are effective solution methods based on the Conjugate Gradient method (for definite problems) or minres (for indefinite problems) in combination with an appropriate preconditioner, which is required in almost all cases. For nonsymmetric systems there are two principal lines of attack: the use of a nonsymmetric iterative method such as gmres, or tranformation into a symmetric problem via the normal equations. In either case, an appropriate preconditioner is generally required. We consider the possibilities here, particularly the idea of preconditioning the normal equations via approximations to the original nonsymmetric matrix. We highlight dangers that readily arise in this approach. Our comments also apply in the context of linear least squares problems as we will explain.

In this talk I will discuss a number of topics at the interface between C* algebras and Geometric Group Theory, with an emphasis on Kazhdan projections, various versions of amenability and their connection to the geometry of groups. This is based on joint work with P. Nowak and J. Mackay.

The goal of this talk is to present some elementary examples of fusion 2-categories whilst doing as little higher category theory as possible. More precisely, it turns out that up to a canonical completion operation, certain higher fusion categories are entirely described by their maximal subspaces. I will briefly motivate this completion operation in the 1-categorical case, and go on to explain why working with spaces is good enough in this particular case. Then, we will review some fact about $E_n$-algebras, and why they come into the picture. Finally, we will have a look at some small examples arising from finite groups.